jaap - It turns out your suggestion is correct. Interestingly, I think did try that approach when I originally got the problem, but it didn't seem to work then - I guess I got bitten by lack of precision of my tools and drawing skills. Since your post, I tested the idea algebraically (with a specific set of circles) and found that the tangent did go through the intersection point.
Another interesting thing is that I do NOT remember enough geometry to be able to solve the problem of "draw a tangent to 1 circle, going through a given point", and I'm not at all sure I ever did.
greengiant - thank you for URL for a full solution. I don't think I could ever come up with that on my own. For anybody who wants to solution without following the link, here's my write up of how to get one of the tangents:
0) Draw your circles and mark their centers
1) Determine the difference between the two circles radii. (Call it D)
2) Draw a circle of radius D around the center of the larger circle
3) Draw a line between two circle centers
4) Determine the midpoint of the line (Call it O)
5) At O, draw a circle that goes through the center of each starting circle
6) Mark points that step 2 circle intersects with step 5 circle
7) Draw line from one of those points to center of smaller circle
8) Draw line from same point that is perpendicular to step 7 line
9) Mark where step 8 line intersects larger circle
10) At step 9 point, draw line perpendicular to step 8 line.
(The Java applet at the link covers how to get all 4 of the possible tangents.)
As for the discussion on what is allowable with straight edge and compass problems, I look at it as there is certain beauty/elegance to defining a system of rules and determining what is and is not possible within that system. For the straight edge and compass set of problems, you can do lots of neat things - duplicate angles, lengths, make parallel lines, make perpendicular lines, bisect angles and lines, and yet, there are things that are impossible - trisect an arbitrary angle, double a cube, square a circle. In college, I did a second semester of abstract algebra and near the end of semester we were able to prove the impossibility of those constructions. I don't remember any details of the proof, but I remember finding it very, very elegant and very interesting.
Thank you again for helping with a problem that has bothered me for years!